On Parent-Child Math Relationships

When I think back to high school, I don’t recall particularly negative things about my relationship with my parents around math class. Up until my senior year, I figured out how to get by (even being a lousy student) with top grades in my math classes. I often received grades I didn’t earn, but because those grades were good, my parents stayed off of my back around math.

Chemistry was a different matter entirely. I took honors chemistry in my 10th grade year, and my father was a research chemist at an elite university. Maybe partly because of that background, this became my hardest class. It was also the one that brought out the worst of both my father and I, and ended up straining our relationship for the next several years.

It was just assumed that I would live up to my father’s reputation by students around me. I’m not sure exactly what my teacher thought of me, or my father, but I always had this impression that he probably thought of my father as an elite Harvard snooty scientist, and probably looked down on me for that. Who knows – maybe I was wrong then, and this teacher didn’t even notice or think twice about me. (He was also the coach of the state champion girls’ basketball team, and I always felt like that came before teaching chemistry for him).

Nonetheless, my home life was pretty difficult that year. There was a regular cycle of my father yelling, me cowering, more yelling, tears, shame, and stubbornness. For nine long months, we fought and fought and fought. It didn’t make me study better. It didn’t make me try to appease my father’s wishes, or spend more time on the class. If anything, it gave me more incentive than ever to not do what he wanted. After all, what did he know about me? About the life of a 14 year old high school student?

I really don’t remember what it was, but towards the end of the year, I did turn things around in my approach, and surprised my father by acing the final. (In fact, he ended up going in and arguing with the teacher about something that got marked wrong and shouldn’t have.) And yet, I wasn’t at a point where I could be grateful for his actions. It took a long time to accept what he had done as an act of caring. In fact, all of the fights that we had came from a place of caring, but when caring interacts with conflict, and especially when there is a passion, it can become explosive.

It took several years before my father and I fully reconciled, but by the time I reached my mid-twenties, he had become my closest friend. I think back to a quote he often recited (and apparently was incorrect in attributing it to Mark Twain):

“When I was a boy of fourteen, my father was so ignorant I could hardly stand to have the old man around. But when I got to be twenty-one, I was astonished at how much he had learned in seven years.”

The last ten years of my father’s life were the ones where we were able to share a lot. We traveled together, backpacked together, and I even got to teach him number theory by going through Davenport’s The Higher Arithmetic. I often wonder how things would have been different between us if we had both reacted differently during that year I took chemistry.

DSCN1500
My father and I, backpacking through the Austrian Alps in 2005.

I used to think that my story was not all that common, before I started teaching. Then I started hearing some stories about the moments that so many students had when sitting at the dining room table, suffering through a homework assignment, with frustrated parents. The mathographies that I had students write earlier this year highlighted some of the rifts that open between kids and their parents over math class. They also confirmed for me something that I’ve often told students. While I love math, and I want them to love math, relationships are not worth losing over math class and homework. This isn’t to say that parents should stop caring, or stop asking, or even stop offering to help with math. It just means that help can mean working with the school community to lift a student up and support that student in their education. With two daughters of my own, now, this is a lesson I have to remember as they continue to grow and go through their own schooling path.

I often tell my students that there are some things more important than math class, and I’ll listen if something outside the classroom is affecting their ability to succeed in my class. This year, I had a similar message for parents at back to school night. Relationships between parents and their teenager kids are naturally difficult. Adolescents are going through many changes, discovering their independence and their boundaries all over again, testing adulthood, and lots of other things that are very hard for parents to grapple with. Adding fights over math and math homework to the mix just isn’t worth the possible repercussions.

Tonight, as I anticipate a World Series with the Red Sox coming up next week, I do have great memories of during in the Fenway bleachers with my father as he taught me how to pencil in every strikeout and hit and run in the scorecard. I have memories of sitting in the university cafeteria, playing with math puzzles on paper napkins (with pen of course).  Those are the kinds of math memories worth holding on to.

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Math Learners Have Stories to Share!

At the beginning of every year for the past several years, I’ve started with simple questionnaires for students, asking about their past experiences in math class. It’s always been nice, but a lot of students didn’t take it all that seriously, and to be honest, since I was trying to get going on content, I didn’t take nearly as much time as I would have liked to make them meaningful. To be honest, it was a self-congratulatory way to make it look like I was getting to know students, without doing a very good job of it.

This past spring, at the NCTM annual conference, I attended a session by Wendy Menard and Emma Gargroetzi  about Mathography Writing. This was just the enhancement that I was looking for, and I was excited to start this year off with student mathographies. The first day, students still filled out a similar questionnaire, but we then used that as a launching point to spend time over the next two classes* telling our stories. (Yes, I told my own story as well). On the third day of class, we also took time to share something that we had reflected on through these three days about our experiences in the past with math and math class.

The first thing I noticed when I started reading through these papers is that they were long. I mean, not pages and pages and pages, but they all really told a story. Some of the stories were exhilarating, and some were heartbreaking. Many students wrote about crying in class, crying in the bathroom at school during class, crying in their bedroom at home, crying at their kitchen table, crying with friends, crying with family. Many talked about feeling dumb or stupid or lazy – words that others used towards them and that they internalized. It wasn’t all bad, though. Most students wrote about a positive experience or two. Some of those good feelings came from good test grades or good class grades, but more came from feelings of accomplishment after solving a problem, or from feelings of wonder at being introduced to some new and surprising idea in mathematics. With all of these stories, I really did start to actually connect my students to their individual experiences as members of the math world. I knew which of my students loved juggling, and which one listens to Iannis Xenakis, and who has a mathematician for a father, and what that means to her. I learned about the student with dyslexia who was pulled from her math classes from 2nd through 5th grade to work on her reading skills, and then was stuck in remedial math in middle school because she had missed so much math class that she was years behind.

The next week, I was approached by multiple students about these mathographies. It was the first time that many of them had ever taken the time to reflect on their math experience as a whole. For some of them, it was a time for realizing that many of their struggles came from external sources, and not just from within. For most, this was the first time that a math teacher had asked them to share these stories, and the first time they felt heard. I hope I can hold onto these stories throughout the year. We are still in the honeymoon period when we are just getting into the content, and our first assessments are coming up later this week. Still, I’m cautiously optimistic. This is the most success I think I’ve ever had in building relationships with students from day one, and the most success I ever have (which is no surprise to most teachers I’m sure) is when I am able to build a connection with the learners in my classes.

I shared one of my highlights in a tweet last week:

Tomorrow’s the next time math club meets since this student approached me, and she’s asked multiple times to confirm. I hope our exploration of Pascal’s triangle lives up to her expectations! (Also, just heard that Pascal wasn’t the first to play with this triangle, so if I get time, I really need to look into that. Ha…time. I don’t really have time to write this, but I also didn’t have a choice – it had to be written!

*During those classes, we did break away from writing to do a variety of math based games – the four 4’s problem was a favorite. We also created class soundtracks. Every student has a theme song, so my Spotify class playlist on shuffle becomes a great way to call random students to share their learning. I think I have Ed Campos, Jr. and Matt Vaudrey to thank for those ideas, but I’m sure others have given me ideas as well.

Thoughts on Equity in MathEd 2018

Diversity used to be the buzzword. Now the buzzword is equity. Both are good words and have great intentions, but there is a danger that they will start to lose some luster as they crop up in less sincere places. Some say we live in a post-racial society, but any reasonable study of the news over the last ten years makes it clear that racism (and many other –isms) are in full force in modern American society. In my bubble of educators in the San Francisco Bay Area, I am surrounded by people who are absolutely not racist, but we all have biases that influence how we teach, and how we interact with our peers and the learners in our classrooms. Yet, despite all of our best intentions, on average, our black and brown students do not succeed in the same way that our white and Asian background students do. In the past, many of us have fallen back on the same excuses that are clearly beyond our control – they did not get the fundamental skills in previous grades, they have parents who do not value education or do not know how to help or are not around to support their kids, they have much more stress at home due to poverty and violence, they don’t have the materials to do work or the tutoring that other kids get. We can even start to scratch at a real issue – we do not have enough teachers or administrators of color in our school who can act as mentors and role models for our black and brown students. It is time for us, as teachers of students of color, to start to recognize that we are complicit in the separate and non-equal education that exists in this country.

I often talk about the question I asked to prospective candidates when we were interviewing for a new head of school. I believe in my school, Mid-Peninsula High School. It is a private school, which raises some concerns for me; whom it is available for could add to the separate but non-equal system, and we may detract from the funding and resources of a public school. That said, we do make some good efforts to be a diverse school, with intentions of a mix of types of people among the student body, faculty, staff, and administration. Efforts and intentions are not enough, though. We have a very white faculty and staff, a largely white administration, and a fairly white student body. We appear to be doing everything right to be welcoming, but the diversity does not happen. So when our previous head was retiring and we were seeking a new head, my question to the three finalists was the same: “We have a very strong faculty and staff, but it is very white. We also have very low turnover. What can be done to increase the diversity of our faculty and staff without simply getting rid of some of our highly qualified teachers?” To be honest, I don’t know, two years later, if there is a good solution for this. I’m not sure there is one, though the notion of “getting rid of some of our highly qualified teachers” does scream to me now of privilege. What I do know is that, of our three candidates, two basically brushed off the question as not really a problem, because we would push for diversity as we hire in the future. That seemed as much of a non-answer as one could possibly give. Only one of them did take the time to emphasize that this is an important question, and one without easy answers, but one which needs to be addressed in the school as we go forward. He was the only non-white candidate, and I am glad to say that he was the candidate that we ended up hiring.

This ties closely to something that I have seen multiple times at diversity workshops or sessions in larger conferences. In every anti-racism session or presentation that I have attended or heard about, it may not be surprising, but there are often people of color either presenting or attending. What has started to strike me recently has been the inherent inequity in this situation. People of color often feel a desire or need to attend sessions regarding equity specifically because racism affects them so directly. A smaller proportion of white people feel the need to prioritize these sessions over other more content-specific or pedagogy-specific sessions. The big injustices that I used to focus on were how unfair it is for people of color to feel a need to always be the ones speaking about racism, and how much the white people who never attended sessions on racism are exactly the ones who need to be better educated about how it affects their own interactions with peers and students and their teaching practices. However, I recently realized something else (which I’m sure has been noted and written about long ago by others who have been involved far longer than I). Every time a person of color prioritizes a session on racism or equity so that they can contribute to the greater good, they are also missing out on a session about pedagogy or content that can improve their own teaching practice. They end up sacrificing for the greater good, and as a result, they are giving up some potentially game-changing professional development. Why are they put in this situation? What can we be doing to improve their experiences?

I recently returned from Twitter Math Camp, a grassroots organization which prides itself (rightly so, I think) on intending to be a welcoming community of math educators. “Intending to be” is the key, though. Despite a large proportion of the population who wants a diverse group of attendees, the vast majority of the people to attended were white. There are a number of reasons why this may happen, but from talking and reading to others, one thing keeps coming up. It is really, really hard to voluntarily be different, even in a welcoming community. It takes a great amount of strength to decide that even though the majority of the people who are attending a conference of white, and the majority of the board is white, it is worth being one of the only non-white people attending, and maybe even the only one you know. I spoke with a few white teachers who have attended TMC in the past who decided not to attend this past year, and in one case it was explicitly in order to hopefully open up the pool of attendees to more non-white people. (A couple of others implied that that was part of the reason). This is noble, and I know that there have been limits on the number of attendees (although that adds to some exclusivity). Again, do we start denying people who are valuable contributors to the math education community, simply because they are white and are taking up space that could be taken by non-white educators? Then again, how much privilege is being invoked in that question right there? Why not deny some white people, maybe especially people who already have a lot of influence and power in the community, and pass some of that power and influence on to others who have not had the same privileged opportunities?

These are difficult questions, I know, and I don’t pretend to have the answers (yet?). I do want to partner with others, though, and find solutions that will better our whole community, from the experiences of the learners in our classrooms to the experiences of teachers and administrators in our schools. Chipping away at racism is a long and difficult task, but is no less important now than it was 50 years ago, 75 years ago, 100 years ago, 150 years ago, 200 years ago, or 500 years ago. It wears different masks through each generation, but its hold on our society is deep and will take an incredible amount of hard work and a lot of time to extinguish. There is a lot of work still to be done in the field of math education, in education in general, and in our greater society before I want to hear anyone utter the phrase “post-racial” ever again.

There are many leaders in the area of equity in math education that I’d like to acknowledge, as following them through their Twitter accounts and blogs and other writings has been so influential to my thinking. I also look forward to them being able to be leaders in other areas of math education as well, as I know they have so much to offer that we don’t often see. Thanks to José Luis Vilson, Marian Dingle, Rochelle GuitierrezRobert Berry (who has, in theory at least, reached the pinnacle of influence in math education), Nicole M. Joseph, and so many more. I have learned so much from each of you about equity, but know that I need to grow and see you as more than that. I hope to see you in the same way I see my friend Lybrya Kebreab, as one who loves and plays with math just as much as one who embraces the importance of social justice and progress in our field.

 

Re-Imagining High School Math

First, I need to start with how much I love the school where I teach. We have amazing autonomy, a truly learner-centered program, where the needs of our students are our highest priority. That said, there are certain things I would change if I could – and to a certain degree, I have earned the trust of my administration to research and explore different ways we can work with our kids to vastly improve their experience and success in mathematics. The school has not had any significant overhaul in the math program, probably since its inception. It is a very traditional curriculum for the most part, with a sequence of Algebra 1 – Geometry – Algebra 2 – Pre-Calculus – Calculus. We also offer an Intro to Algebra class and a Consumer Math (now Financial Literacy) class, for students who wanted or were perceived to need an alternative to the traditional sequencing in order to fulfill their requirement of 3 years of math. As a result, we have students who enter our school at a large variety of math background, and leave with almost as much variety in the math they have seen over their four years in high school. There is no real change in equity – most Black and Latinx students do not take calculus, most students with learning disabilities do not take calculus, and the calculus class is usually mostly white or Asian students. Furthermore, students who do well only have the option of calculus, even if they do not plan to pursue studies in fields that will actively use calculus techniques and would benefit from other class options.

I’ve been thinking about how to better serve our students for a while now, and this past April at NCTM Annual Meeting in Washington, D.C., I picked up a copy of the highly promoted book Catalyzing Change in High School Mathematics, which gave me a lot of the framework and research that I was looking for, all compiled in one place. Between that and the “Re-Imagining High School Mathematics” morning session at this summer’s Twitter Math Camp, led by Carl Oliver and Sadie Estrella, I was able to work through a long term plan outline. There are a lot of details to work out, but here is the general framework of what I want our students to have for their classes:

  • Move from a traditional Algebra 1 – Geometry – Algebra 2 -Pre-Calculus sequence to an integrated Math 1 – Math 2 – Math 3 sequence that incorporates the most essential mathematical skills that students will need.
  • Work with the other math teacher(s) and administrators to determine what those essential mathematical skills are. These are the skills that we expect all high school graduates to have in order to be well prepared for any further math class, as well as to participate fully and successfully in society.
    • Algebraic reasoning skills
    • Financial literacy
    • Understanding of plane and solid geometry, measurement, etc.
    • Ability to read, interpret, and evaluate data, graphs, and basic statistics
    • Preparation for calculus, statistics,  discrete math, or computer science classes
  • Develop a support class, preferably a zero period class that can be required for some students and a drop-in class for others.
    • Give students a just-in-time support for reviewing essential skills before they come up in the core Math 1/2/3 classes
    • Extra help, support, and practice for the content of the core classes
    • Opportunity for peer tutors to receive credit as well as review and build leadership skills
  • Create multiple options for a fourth year class, including Calculus and Statistics.

There are going to be (very valid) concerns about moving to this model. Firstly, many of our students with various learning disabilities may feel overwhelmed by facing math that may be familiar to their peers. Secondly, many students who took Algebra 1 in 8th grade may feel that they will not get much out of a class with so much review (in their eyes). Thirdly, many students who need that support class may have a hard time arriving on time if it’s a zero period class that starts 45 or 50 minutes before their first class. (This is especially true of students with anxiety who may avoid attending school when overwhelmed).  Fourthly, will we be able to handle such significant changes in terms of staffing and scheduling, and will we be able to hire a third teacher? It is going to be up to us to see what we can do to address or alleviate these concerns, and will take some good research and planning to do so.

Despite the challenges we have ahead, and the many questions we have to put forth and then answer, I’m looking forward to bringing this proposal formally to my department and administration this fall. A change like this won’t happen immediately, and I don’t foresee any change at all this year. I’d like to put the plans into place over the coming year to start rolling out a new set of classes, starting with an integrated Math 1 and Support Class in the 2019-2020 to replace our Intro to Algebra and Algebra 1 classes. By the 2022-2023 school year, my hope is to have a math program where (except in extraordinary circumstances) all students are taking Math 1-3 in 9th-11th grades, and taking a fourth year math elective in 12th grade. My goal is to see Calculus and Statistics classes that are full and reflect the ethnic, racial, LD, gender, and other diversities that make our school so great.

Late #TMC18 Recap

I got back from Cleveland nine days ago. After my wife’s quick business trip to Chicago, my older daughter’s 7th birthday party, and my mother-in-law arriving for a visit, I am finally getting an opportunity to reflect on my second time at Twitter Math Camp. There is something amazing to me about getting together with passionate math education professionals, many of whom pay their own way to this informal, grassroots conference. The NCTM Annual Meeting has its benefits, with enormous vendor areas, nationally known keynote speakers, and exciting new locations. My local state conference, CMC-North at Asilomar, is much more intimate, in a beautiful oceanside resort setting, with outstanding presenters and attendees. TMC is a mix of both worlds, with a new location each year, an international crowd, and small groups working on interesting things.

This year, my speaking proposal (on matrices and matrix applications for beginners) wasn’t accepted, but I was able to get in through the lottery. In some ways, I started to feel a bit of impostor syndrome sinking in, since the whole point of TMC is really to encourage everyone to share and collaborate. That feeling didn’t really leave until partway through the conference, when Julie Ruelbach gave a great lunchtime keynote on how prevalent impostor syndrome actually is, and reminded us all that we do belong. At that point, I thought about submitting a #MyFavorites short presentation, but wasn’t convinced I’d be able to put something together in time.  Next time I go, though, I’ll be sure to stand up and speak. On what? A debate format perhaps? Rethinking homework? How I do honors classes? Maybe something about 3D printing? Maybe a bit of linear algebra after all? How about the role of white teachers as allies for students of color, and how we can press forward with the hard questions surrounding equity? There are plenty of things I can speak about, many topics in math and education that interest me, and surely I have something to offer after 17 years in the classroom!

On to the sessions, with a brief description. I’ll be writing some follow-up posts about a couple of the big ones.

Morning session: Reimagining High School Math with Carl Oliver and Sadie Estrella. This session took place over three consecutive mornings for two hours each morning, and was an opportunity for teachers and teacher leaders to think about a way to change their own classroom, school, or district math program. I am fortunate to teach in a school with amazing support, and have received encouragement to work on my plan to revise the math program at our school. We have an interesting and diverse population: many students with diagnosed learning disabilities; many students with high anxiety around school in general and especially with math; several students who come from schools in lower-income areas with fewer resources, where their transcript that shows high grades and Algebra 1 in 8th grade may not match with a mathematical background that has not exposed them to deep and rich mathematical thinking. The question I have had for a long time is how to ensure that all students are given an opportunity to thrive in mathematics to the absolute best of their ability, regardless of their previous experiences. Other teachers in the session had a variety of goals and plans, but all related to improving the experience of math learners in their mathematical appreciation and ability. (Even Chris Nho‘s plan to improve the community feel of math educators in his district has potential long term benefits of building camaraderie and then collaboration among teachers in his district.) Check the #rehsmath hashtag on twitter to see more!

Afternoon sessions:

Promoting Equity through Math Talks, with Anne Agostinelli. Thinking about how math talks can be designed to progress throughout the week, with intentional structure, in order to address the SMP‘s consistently.

Interactive Notebooks for All, with Farica Erwin. (I’ve submitted to speak with Farica at the NCTM Annual Meeting in San Diego on rethinking homework, and this was the first time we met in person!) I’m not big into doing lots of crafty things, but I know that lots of people, teachers and learners alike, love interactive notebooks. What did I take away? The importance of being intentional about the interactivity, of not overdoing it, and most importantly, of page numbers and a table of contents. In fact, that’s the biggest takeaway, as I think about how I may try out digital notebooks this year in my classes – organizing by a table of contents and shared/common vocabulary so that all students can have a structure in place.

Living Proof: Enjoying Teaching 2-Column Proofs, with Elissa Miller. I may not be crafty, and Elissa is incredibly crafty, but I took away some great ideas and resources in this session. Lots of the proofs presented reminded me of the fun and creative ones found in Harold Jacobs’ Geometry (now out of print, unfortunately).

Sausages Without The Skin, with Nik Doran and Max Ray-Riek. I’ve been following the work of Illustrative Mathematics for a little while now, and have watched as they have snatched up so many great teachers to work with them on their new curriculum. Now they are getting ready to release new high school resources, and this was a chance to take a sneak peek at what they are creating. Although I’m hoping to move towards an integrated curriculum, which isn’t what they are currently working on, I do expect to be able to incorporate a lot of their lessons into our curriculum as we move forward over the next several years.

Anxiety, Mindset and Motivation: Bridging from research to practical classroom
structures, with Lisa Bejarano and Dylan Kane. I was really excited to see what Lisa and Dylan are doing to bring the research based ideas of Jo Boaler, Carol Dweck, and Ilani Horn directly to the classroom. We came away with some great ideas, including a new routine for my next year, Sara Van der Werf’s Stand and Talks.

Flex Session: Equity at TMC, with Tina Cardone and Sam Shah. This was a continuation of a session from last year, and one that has been on my mind for a while now – not just at TMC. Math education is a place where so many teachers are white, and so many visible leaders are white, and conferences tend to skew even whiter than the population of math teachers. If we want students of color to thrive, we need to ensure that they have teachers of color. Furthermore, if we want white students to buy into the capabilities of their black and brown classmates, they need to have teachers of color as examples and role models. (That’s not all we need, clearly – we have many math teachers who are women, but there are still fewer girls than boys going into math as college majors. Still, increasing the number of women who are teachers is an important step.)

Besides those great sessions, there were so many impromptu conversations, great socializing with new and old friends (including a vegan contingent), and a chance to see Cleveland, a place I’d never have intentionally sought out but which turned out to be a wonderful and welcoming city.

Next year it’ll be in Berkeley, and I hope to be there again, seeing friends and sharing in this great profession that we have chosen.

Algebra Review: Area of a Trapezoid

There are lots of nice and simple and very visual derivations of the area of a trapezoid. Those are the ones I hope my students hang on to and rely on when they are thinking about how to find and calculate its area. (We’ve already talked about my reservations with simply memorizing area formulas, and the problems that can arise from rote memorization).

However, in my geometry classes, I like to present this algebraic derivation of the area formula for a trapezoid. This is about the point in the year where students have forgotten a lot of their algebra skills from last year, and even cycling through some of those skills in classwork and homework only does so much. Playing around with some ideas, including thinking about a trapezoid as a decapitated triangle (because that’s clearly what it is, right?) led me to this derivation. I won’t claim that I’m the first to come up with it, and searching around online I’ve found some similar derivations, but I really like this one because of all the ideas it touches on: similar triangles, somewhat complicated proportional relationships, distribution and factoring of algebraic expressions, manipulating fractions, and more. Even though a few eyes start to glaze over when I first tell students I’m going to lead them through a derivation filled with lots of algebra, they start to get interested when they see that there are places where those algebra skills can really help make sense of a situation.

Here’s the derivation:

Deriving the Area of a Trapezoid (With Lots of Algebra Review)

Suppose we have trapezoid \square BCED with an altitude of length h drawn through points F on \overleftrightarrow{BC} and G on \overleftrightarrow{DE}. Let BC=b_1 and let DE=b_2.

Trapezoid BCDE

Extend the legs until they intersect at point A, creating \triangle ADE and \triangle ABC. Since \overline{BC} \parallel \overline{DE}, we know that \triangle ABC \thicksim \triangle ADE. Also, label the intersection of \overline{DE} and the line perpendicular to \overline{DE} through A as G, and label the intersection of \overline{BC} and \overline{AG} as F. Then FG = h is the altitude of trapezoid \square BCED, and the altitude of  \triangle ABC is AF=h_2.

Since we have similar triangles, we can write the following proportion: \frac{h_2}{h}=\frac{b_1}{b_2 - b_1}. Also, since \alpha \triangle ABC + \alpha \square BCED = \alpha \triangle ADE, we can subtract to find the area of the trapezoid: \alpha \square BCED = \alpha \triangle ADE - \alpha \triangle ABC. The areas of the triangles can be calculated: \alpha \triangle ABC = \frac{1}{2} (b_1)(h_2) and \alpha \triangle ADE = \frac{1}{2} (b_2)(h_2 + h). When we substitute the area calculations, we find that:

\alpha \square BCED =  \frac{1}{2} \big[(b_2)(h_2 + h) - \frac{1}{2} (b_1)(h_2)\big]

At this point, we have the area of the trapezoid in terms of the measurements of its two bases (b1 and b2), the measurement of its height (h), and the measurement of the height of , h2. We can use the proportion we found, \frac{h_2}{h}=\frac{b_1}{b_2 - b_1}, and solve it for h2 to find  h_2=\frac{b_1 h}{b_2 - b_1}. Substituting this in the equation above gets an expression for the area of trapezoid  in terms of the two bases and the height of the trapezoid.

\alpha \square BCED = \frac{b_2 h + b_2 \bigg(\frac{b_1 h}{b_2 - b_1}\bigg) - b_1 \bigg(\frac{b_1 h}{b_2 - b_1}\bigg)}{2}

Factoring out \frac{1}{2} and simplifying the fractions:

\alpha \square BCED = \frac{1}{2} \bigg( b_2 h + \frac{b_1 b_2 h}{b_2 - b_1} - \frac{(b_1)^2 h}{b_2 - b_1} \bigg)

Then multiply b_2 h by \frac {b_2 - b_1}{b_2 - b_1} to make all the terms inside the parentheses have the same denominator:

\alpha \square BCED = \frac{1}{2} \bigg( b_2 h \bigg( \frac {b_2 - b_1}{b_2 - b_1} \bigg ) + \frac{b_1 b_2 h}{b_2 - b_1} - \frac{(b_1)^2 h}{b_2 - b_1} \bigg)

Now distribute and place everything inside the parentheses into one fraction and simplify:

\alpha \square BCED = \frac {1}{2} \bigg( \frac {(b_2)^2h - b_1 b_2 h + b_1 b_2 h - (b_1)^2h}{b_2 - b_1} \bigg)

\alpha \square BCED = \frac {1}{2} \bigg( \frac {(b_2)^2h - (b_1)^2h}{b_2 - b_1} \bigg)

Factoring out the h leaves:

\alpha \square BCED = \frac {h}{2} \bigg( \frac {(b_2)^2 - (b_1)^2}{b_2 - b_1} \bigg)

Then factoring the difference of squares gives:

\alpha \square BCED = \frac {h}{2} \bigg( \frac {(b_2 + b_1)(b_2 - b_1)}{b_2 - b_1} \bigg)

Now divide out the b_2 - b_1 and rearrange to get our familiar area formula:

\alpha \square BCED = \frac {h (b_1 + b_2)}{2}

There are a lot of things I like about this approach, but the engagement of students as I lead them through this proof is a lot of fun. I rarely stand up and lecture or give students almost everything in a proof, but there are times when lecturing is effective and shouldn’t be summarily abandoned. Plus, this also gives me a chance, as we go through each step, to walk around and see which students may need more help with fractions or factoring as we continue through the year, and I can give them some additional help and resources. It’s also an opportunity to talk with the Algebra 2 teacher about the upcoming class and their current skill levels with important concepts.

 

 

A Curiousity in Geometry

One thing that I try to emphasize to my students is that mathematicians don’t spend their time (usually) doing calculations. They spend a lot of time trying to make sense of scenarios, and trying to answer bigger questions by breaking them down, and they spend a lot of time being curious. One of the most important skills of a mathematician, to me, is to try to find a good question. Last week, we had one of my favorite geometry lessons of the year, in which we just spend the entire class playing and posing questions.

I asked students to spend the first 10 minutes of class just constructing isosceles triangles, and whatever may go along with them. At this point in the year, we have covered constructions of perpendicular bisectors, midpoints, angle bisectors, and we just discovered a variety of triangle centers: orthocenters, centroids, incenters, and more. Students are encouraged to use either pencil and paper, GeoGebra, or both. After playing around and having several sketches and constructions completed, I encourage them to spend 5 minutes looking at their own drawings as well as those of the other three group members, and do a little noticing and wondering. While they do this, I model this process as well, both with a doc camera and with GeoGebra on my SmartBoard.

I’m always impressed with the variety of questions that students come up with, and this year was no different:

Isosceles Questions P1Isosceles Questions P5

Check out how creative these students are! How curious! How thoughtful! How perceptive! I encouraged them to try to find questions that they don’t know the answers to, and then challenged them to find questions that they think I don’t know the answers to.

I’d also like to share something that I found, when playing around with GeoGebra. I constructed an isosceles triangle based on a circle, and then constructed the orthocenter, and then animated one of the legs as I traced the path of the orthocenter.

Orthocenter-Animation.opt.gif

I made a video and narrated it, showing the steps as well.

Well, it turns out that when you do interesting things and share them on Twitter, you get a wonderful variety of interesting other things happening.

Mark Kaercher (@shskaercher) had all sorts of brilliant ideas to explore:

Steve Phelps (@giohio) suggested tracing the incenters and excenters together, and later put a whole bunch together:

Henri Piccioto (@hpicciotto) modeled my original question in Cabri:

And then Henri explored further:

I was thrilled to watch this thread, although I barely was able to take part. You see, I started this post on Monday morning before school, and partway through my second class on Monday, a family emergency came up and I’ve been otherwise occupied for most of the time since then. For anyone I haven’t told who reads this and is curious, please feel free to reach out and ask. Everything isn’t great or okay, but I’m being cautiously optimistic at the moment, and really yearning to get back to the classroom and back to the world of math and teaching and a sense of normalcy.

List of Blog Posts to Write

This year, I was so excited. I have a prep period, I stopped requiring (and grading and commenting on) homework assignments, and suddenly had so much free time and no excuse to not write. Except that sometimes, life happens. This wasn’t a case of anything catastrophic, or even bad. In fact, a lot of it was good. I started attending math teacher circles each month. I spend more time with my wife and daughters at night and on weekends. I am getting more sleep (except when our three year old decides to crawl into our bed at 3AM, kick me, and take my covers). After Carl Oliver’s great blog post, and follow up presentation at Twitter Math Camp to #PushSend, I really wanted to write more regularly, whenever thoughts came into my head. And yet here it is, just before Thanksgiving, and I haven’t written anything since the beginning of the school year.

It’s not like there haven’t been a whole lot of things on my mind. Maybe having a list of things that I wanted to write about will help me make it happen. So here it is…the list of phantom blog posts:

  • Thoughts on my math students who have received cultural messages that they are not cut out to do math, whether due to disability, due to ethnicity, due to race, due to gender, due to speed, due to athleticism, or other factors that I may not be aware of, and what I can be doing better.
  • Thinking about a quadratic equation as both a product of lines and as a sum of a quadratic, linear, and constant term, and how to think more geometrically about algebra.
  • Development of formal debate in my math classes, including the technology that we use to make debates accessible to all.
  • That non-binary student who is often mis-gendered, and my uncertainty about whether I have taken the right approach each time.
  • Reasons that I teach proof at the beginning of Geometry (which isn’t a novel sequencing, but one that I’ve thought a lot about).
  • Experiences with not requiring homework in most cases, including in-class homework reflections, increases in how much homework is done (admittedly through self-reporting from students), and decreases in stress and workload for students and for me.
  • Year 2 of my implementation of standards based grading (in a school that uses a traditional grading system on report cards), and what I learned from my first year.
  • The amazing things my math club does every week, largely with ideas mined from #MTBoS (Twitter Math Educator Community).
  • Working through my implementation of a CPM Algebra 1 curriculum, which I largely like but find it difficult to deviate from.
  • The good, the bad, and the ugly of the Barbie Bungee project.
  • The number of very capable Pre-Calculus students who decide not to take on the challenge of doing the class at an honors level, and how I can make it more enticing, and not just seem like more work.
  • The awesomeness of our faculty band.
  • Experiencing Imposter Syndrome in year 17 of teaching.
  • Playing with math games in my core (SET, Wits and Wagers, and Prime Climb are favorites so far).

On a good note, I have continued to be relatively active on Twitter, and should (hopefully) be well prepared for my talk in (eek!) 10 days at Asilomar!

Simplifying with Complexity

Sometimes the quickest and easiest way to solve an easier problem isn’t the best way to approach a hard problem. Today (first day of course content in my Pre-Calculus class) we were reviewing sets of numbers, categories of numbers, and notation. I gave my class the following set:

{-2, -1, 0, 1, 2, 3}

and asked them to turn it into set builder notation. Every student was able to write the set as some variant of the following:

Set Builder AV

I asked if there were other ways of writing it, and some people changed the set of integers to whole numbers or natural numbers, but that was about it.

Then I asked the class to do the same thing with the following set:

{3, 6, 9, 12 … }

And they struggled, and struggled. I saw a lot of students just going nowhere. As they worked, I went back to that original set, and wrote up a more complicated solution:

Set Builder EW

One student looked up, then another. A couple of students started working out how my solution worked. A couple of others asked for clarification. And then, one by one, I started to hear those magical “Ohhh!” sounds that we all love – you know that light bulb moment. Students started to remember that they could match each term to a number, and most of them came up with something like this:

Set Builder RBFrom a desk off to the side, I heard a student exclaim, “Whoah! When you showed us a more complicated way to do the easy problem, it made it easy to do the hard problems!”

 

Day 1 Debate – What’s the Best Number?

I know that by now, most schools have started. I think I’m one of the last that still starts after Labor Day – not that I’m complaining. A popular blog post out there is a first day activity, and I wanted to share the one that I’ve done the past few years. Yes, it’s a debate, but a totally informal debate, one that works at any level of mathematical background, students have a lot of fun with, that builds both competition and teamwork in a low stress way, and tells me a surprising amount about my students.

After normal introductions, I give students a simple task – to come up with a number. It could be a favorite number, or a really interesting number, or a number that has some personal meaning. I then ask them, once they’ve decided on a number, to come up with as many interesting things about that number as they can, and give them a couple of minutes. They can use calculators, they can use the Internet, they can draw, and if they get stuck they can ask me for help (though, to be honest, they rarely ask for help with this).

After a few minutes, when at least some of the students are feeling like they’re done, I have them get into pairs, and then in the pair they decide which number is better. They are given about one minute to make their cases and decide, and once each pair has decided, I have each pair find another pair, and decide which of the two numbers is better. It’s interesting to me that I never actually describe this to students as a debate, and in theory they are working together, but they do have something invested in the number that they came with. Inevitably, groups start to argue, but generally nicely, and the whole idea of comparing the best and worst qualities of numbers becomes a source of passion.

The process of finding another group and then discussing, then finding another group and discussing, continues until you have (hopefully) two halves of the class shouting at each other about whether 32 is a better number than 360 (because powers of 2 are more important than having lots of factors and describing the degrees of a circle), or whether 12 is better than 18 (but of course 12 is better). bradymanning

In listening to conversations that happen, I can get to know an amazing amount about student interests, as well as which students feel very comfortable with what numbers mean and how they can be manipulated and described mathematically. On top of it all, having an entire class passionately engaged in a meaningless debate about which number is best, where you can catch every student having fun playing with math from day one, is a pretty great way to start the year in my opinion.

So, what’s your favorite number? Why?